Propagation constants of a fluid-loaded periodic beam using FEM

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15 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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1

1
Propagation constants of a fluid
-
loaded periodic beam
using FEM


Abhijit Sarkar, Venkata. R. Sonti

Facility of Research in Technical Acoustics

Department of Mechanical Engineering

Indian Institute of Science, Bangalore
-

560 012


Abstract

The finite ele
ment method is used to compute the propagation constants for a fluid loaded
infinite periodically supported beam. At most frequencies there is at least a single
propagating wave which corresponds to the acoustical pressure. At other frequencies
there exist

two or even more propagating waves because of the presence of coupling
between multiple domains. The coupled natural frequencies of the periodic unit are
closely associated with the critical points on the propagation constant curve.


1

Introduction

Railw
ay lines, ship hulls, multi
-
span bridges, aircraft fuselages are periodic structures
where a basic unit (span or period) is repeated throughout the structure. Conveniently,
one needs only to model the basic unit and impose periodicity conditions to obtain
the
system equations for the whole structure. There is extensive literature on periodic
structures, a comprehensive review of which is presented in [1]. Although fluid loading
has been investigated in the context of sound radiation from periodic structures

[2, 3],
most of the related literature has struggled with semi
-
analytical and semi
-
numerical
methods and attempted mostly 1
-
D problems. With a final goal of attempting a large
scale real world periodic structure problem (ship hulls and submarines) using o
nly
numerics (FEM/BEM), in this paper we present a preliminary study of a simple fluid
-
loaded infinite periodically supported beam.

The specific system considered involves an infinite beam with periodic supports. The
beam is loaded all along its length wit
h a column of water of height h (see figure 1). We
use the Finite Element Method to formulate the equations and analyze this system.


Figure 1. Schematic of an infinite fluid loaded periodically supported beam.




1

This paper won the student prize at

the National Symposium on Acoustics, 2005, NAL, Bangalore.


2

2

Theory

For an infinite periodically supporte
d beam in free vibration, the displacement in one
span is

e
times that in the neighboring span [4]. The variable


is usually complex and
is called the propagation constant. The real part of


is called the attenuation constant and
the

imaginary part the phase constant. For the system considered, with the fluid loading
neglected, the propagation constant is presented in figure 2. The material of the beam
considered is steel with a Young’s Modulus of
200Gpa
and a density of 7840kg/
3
m
. The
beam has a square cross
-
section of side
0.1m
and rests on simple supports placed at
intervals of
1m.

It was verified that

each propagation band starts from the natural
frequency of the single span simply supported beam and terminates

at the corresponding
clamped
-
clamped natural frequency [4].



Figure 2: Analytical evaluation of propagation constant in the uncoupled case


The FE implementation for structural dynamics is given in any introductory textbook on
finite element me
thod [5]. The FE model in frequency domain (with circular frequency

) leads to the following equation












F
x
K
x
M
s
s



2

. (1)

The FE formulation of the acoustic helmholtz equation [6
] gives a similar equation












v
p
K
p
M
a
a



2
2
c

. (2)

In equations (1) and (2),
M
and
K
denote the mass and stiffness matrices, respectively.
The subscript
s
and
a
refer to the structural and acoustic
quantities.
F
and
v

are the
structural and acoustic load vectors, respectively.
c

is the sonic velocity in the acoustic
medium. In structural acoustics, the acoustic pressures form the structural load, i.e., there
is a relation of the form
{F}
=
[ C ] { p

}
. Also, at the fluid
-
structure interface the
acoustic velocities need to match the structural velocities. This would lead to a relation of
the form
{v} =
2


[S]{x}.

The exact form of the
C
and
S

matrices is as given in [7].

3

Combin
ing these relations with equations (1) and (2) we get the following eigenvalue
problem for the coupled structural acoustic system


































0
0
p
x
M
S
0
M
p
x
K
0
C
-
K
a
s
a
s
2

. (3)

Periodicity conditions are then enforced in equation (3). Orris and P
etyt [8] describes the
methodology for this. As a result we have


dependent stiffness and mass matrices. The
resulting complex eigenvalue problem is










d
M
d
K
)
(
)
(
2




. (4)

Th
e expressions for
)
(

K

and
)
(

M

are given in [8]. Solution of the eigenvalues (

)
for purely imaginary values of


(between
-


and

) gives the propagation bands. Every
imaginary value of


results in n eigenvalues, where n is the system size of equation (4).
Of these n eigenvalues, the i
th

one is a single point
in the i
th

propagation band.


3

Results and Discussion

Using the methodology elaborated, we compute the propagation constants for the fluid
loaded infinite beam shown in figure 1. Figure 3 shows the propagation constant in the
frequency range 0 to
1.1 KHz
.

This result contrasts with the earlier result of uncoupled
analysis as shown in figure 2. There are a few salient features in this result that we wish
to highlight in the following discussion.


The complex eigenvalue problem in equation (4) solved for
imaginary values of


between



and


results in real eigenvalues

. The imaginary part of the calculated
eigenvalues are about
6
10


times the real part. This is in consonance with the physica
l
understanding of the problem. In the absence of any dissipation mechanism in the model,
purely imaginary


should correspond to real frequencies. For an unbounded acoustic
medium with non
-
reflecting boundary conditions at infinity th
e structure would encounter
a radiation damping effect. The imaginary


values will result in complex frequencies
and in turn real frequencies will result in complex


values. There will be no more a pure
propagation band. Future work is planned in this direction. The mathematical reason for
getting real eigenvalues on solvi
ng the complex eigenvalue problem of equation (4) is
however not clear. The observation that imaginary


values result in real eigenvalues for


demands that there be a certain structure in the
M

and
K

matrices.

It may be noted
that due to the coupling terms these matrices are no longer symmetric. However
symmetry is a sufficient condition for getting real eigenvalues and is not a necessary
condition.


As seen in figure 3, for a given frequency, if


is a propagation constant, then
-


is
also a propagation constant. The sign reversal actually implies reversal of propagation
direction of the free waves. From hereon, we focus on the positive propagation constant.

We have c
hosen imaginary values of


in solving equation (4). Thus, our focus is
only on propagation zones for the system considered. It is observed that each frequency
has at least one imaginary propagation constant and some have even more. The existence
of more than one propagation consta
nt for some frequencies seems reasonable as there
are more than one coupling degrees of freedom between the periodic units, namely
-

beam rotation and acoustic pressure [4].


4


Figure 3: Imaginary part of propagation constant for the coupled problem


The ac
oustic pressure propagates at all frequencies. The straight line segment in the
propagation constant curve in the figure between 0 and 750 Hz is indeed the acoustical
dispersion curve (since the span length is 1m). The straight line with the negative slope

beyond 750 Hz occurs because


values are phase wrapped around


and


. As seen
from figure 2, the first propagation band for the uncoupled beam starts at
221Hz

and ends
at
519Hz
. It is noted from figure 3 that within this band there are two propagation
constants. Thus, in this band the energy is carried by both the acoustic pressure as well as
the beam rotation.


The results of the uncoupled analysis in figure 2 showed that the natural
frequencies of the simply supported and clamped
-
clamped periodic u
nit form critical
points of the propagation constant curve. Two more cases have been analyzed with regard
to the single fluid
-
loaded span in figure 1. In the first case, the beam was simply
supported and zero velocity was assumed along the boundary. The fi
rst three natural
frequencies were
229Hz, 682Hz
and

750Hz
. In the second case, the beam was clamped
on both ends and all acoustic pressures at the left and right edges were constrained to
zero. In this case, the first three natural frequencies were
519Hz,

750Hz
and

1014Hz
. The
frequencies corresponding to the former case are presented in figure 3 with bold solid
vertical lines while for the latter case they are in bold dash
-
dot vertical lines. As with the
uncoupled analysis these frequencies form critical
points of the graph of the propagation
constant. Further investigations are underway to understand these results more clearly.



5

4

Conclusion

Propagation constants for a fluid
-
loaded infinite periodic beam were presented using an
FEM formulation. Although
a very simple system was considered, physically meaningful
results were obtained. It was found that there is a propagating acoustical wave at all
frequencies, whose propagation curve matches the acoustical dispersion curve. At some
frequencies there exist
two or more propagating waves which are related to the coupled
fluid
-
structure degrees of freedom.


The future plan would be to extend this analysis for the case of a fluid loaded two
-
dimensional plate. Also it is of greater interest to model an unbounded

acoustic medium
in the present setting. Thus, use of the infinite element method and the boundary element
method could be explored. However, as noted earlier the unbounded acoustic medium
would be equivalent to introducing a dissipation mechanism in the m
odel in the form of
radiation damping. This would complicate the simulation process. Also, practical
structures are never infinite. Structures having a large number of periodic repetitions
approximate the behaviour of infinite periodic structures. There i
s an easy and efficient
method of arriving at the natural frequencies and mode shapes of such large, finite,
periodic structures from the behaviour of its infinite counterpart. This has been covered in
the context of uncoupled analysis by Sengupta [9]. It
would be challenging to arrive at an
analogous method for coupled structural acoustics.


References

[1] D. J. Mead 1996
Journal of Sound and Vibration
190
(3), 495
-
524, Wave propagation
in continuous periodic structures: Research contributions from Southamp
ton, 1964
-
1995.

[2] B. R. Mace 1980
Journal of Sound and Vibration
73
(4), 473
-
486, Periodically
stiffened fluid loaded plates, 1: response to convected harmonic pressure and free wave
propagation.

[3] B. R. Mace 1980
Journal of Sound and Vibration
73
(4),
487
-
503, Periodically
stiffened fluid loaded plates, 1: response to line and point forces

[4] D. J. Mead 1970
Journal of Sound and Vibration
11
(2), 181
-
197, Free wave
propagation in periodically supported infinite beams.

[5] T. R. Chandrupatla and A. D. B
elegundu, Introduction to finite elements in
Engineering, Prentice Hall India.

[6] A. Craggs, Acoustic Modelling: Finite Element methods in
Encyclopedia of Acoustics
edited by Malcolm J. Crocker, John Wiley & sons Inc.

[7] A. Sarkar, V. R. Sonti and R. Pr
atap 2005
International Journal of Acoustics and
Vibration
10
(1), 1
-
17, A coupled FEM
-
BEM formulation in structural acoustics for
imaging material inclusions.

[8] R. M. Orris and M. Petyt 1974
Journal of Sound and Vibration

33
, 223
-
236, A finite
element s
tudy of harmonic wave propagation in periodic structures.

[9] G. Sengupta 1970
Journal of Sound and Vibration
13
(1), 89
-
101, Natural flexural
waves and the normal modes of periodically supported beams and plates.